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Model check
The science
What is simulated
A planar double pendulum built from two rigid links. Each link is a uniform rod of mass mr carrying a point bob of mass mb at its end, so the moments of inertia are J = mr l2/3 + mb l2. Angles are measured from the downward vertical.
Equations of motion
The Lagrangian L = T - V uses T = ½ M11 ω12 + ½ J2 ω22 + c cos(θ1 - θ2) ω1 ω2 and V = -G1 g cosθ1 - G2 g cosθ2, with M11 = J1 + m2 l12, c = l1 S2, where S is the first moment of a link about its pivot. The two Euler-Lagrange equations form a 2 x 2 linear system for the angular accelerations, solved exactly at every step and integrated with fourth-order Runge-Kutta at a fixed 1 ms step.
Small swings and chaos
For small angles the equations become M θ'' + K θ = 0 with two normal modes found from det(K - ω2 M) = 0: a slow in-phase mode and a fast out-of-phase mode. Large swings couple them nonlinearly and the motion is chaotic: a twin pendulum started 1e-6 rad away separates roughly as exp(λ t). The page fits λ to the part of the separation curve between 1e-5 and 0.1 rad, before the gap saturates at the size of the motion.
Simplifications
- Motion is in one plane; the Blender rig is drawn with the two arms in separate layers so they can pass each other.
- Friction is linear viscous torque at the two bearings; real bearings also have dry (Coulomb) friction and air drag grows with speed squared.
- Rods are perfectly rigid and uniform; bob size does not enter the moments of inertia (the bob is a point mass).
- The Lyapunov value is a finite-time estimate from one orbit, not an ensemble average.
Checks
Five boxes: energy conservation over 60 s, agreement with the textbook point-mass closed-form accelerations, small-angle normal-mode frequencies against measured periods, time reversal, and a Benettin Lyapunov exponent for a chaotic and a regular orbit.